Becomes FA = kj =(-1) j1 sinhj 0 j 0 sinh 2(-1)n (k
Becomes FA = kj =(-1) j1 sinhj 0 j 0 sinh two(-1)n (k n)(2 k n)(1 2k) 2 kn n=0 n! ( k ) (2 k ) (1 2k n )(1 z )Y kn dY[sinhj 0- (1 – Y ) sinhj 0 2k n 2 ],(144)where Y = 1 – 2 . The integration with respect to Y might be performed trivially,1Y dY = [ a – b(1 – Y )]21/aX dX a -1- = , a-b ( a – b)(1 )(145)where X = Y/( a – b bY ). This permits the summation more than n in (144) to be performed in terms of yet another hypergeometric series, yielding k 2 (1 k)j 0 -1-2k 2 ) 2 j 0 two j 0 2 k j=1 (sinh 2 – sinh two )(1 z )FA =(-1) j1 sinh j0 (sinh2 F1 k, 1 k; 1 2k; -(1 z two ) -sinhj 0.(146)The argument on the hypergeometric function shows that the limit of higher temperature ( 0 0) can not be taken simultaneously using the limit of higher z. Within the absence of a uniformly asymptotic expansion, we now proceed to derive separately the significant z and also the huge T0 behaviours. In the case of big z, we can treat the argument in the hypergeometric function as a tiny quantity, which permits the series Pinacidil In Vivo representation (A11) to be employed:two Fk, 1 k; 1 2k; -(1 z2 ) sinh2 j0= 1-k (1 k )(1 2k)(1 z2 ) sinhj 0 O[(1 z2 )-2 ]. (147)An expansion with respect to smaller 0 has to be performed so that you can extract the big temperature PK 11195 MedChemExpress behaviour of FA (z). Exchanging the summation over j with all the above series expansions is just not strictly valid and can result in a nonuniform asymptotic expansion. Taking j into account the (sinh 2 0 )-1-2k term appearing in Equation (146), the massive temperature expansion includes each damaging and positive powers of j. Restricting only to terms withSymmetry 2021, 13,30 ofvanishing or negative powers, the summation more than j might be performed making use of Equation (A4), top to (1 k ) 2 two ( 1 k) 1- 2 T0 1z2 2kFA (z) =(2 2k) 1 -1 (2 T0 )two 212k-1 (2k ) 2 – 1 – 2k-1 (3 2k ) O( T0 2 ) 6 two T0 1z2-4k(1 k ) 1 2k(4 2k) 1 -1 (two T0 )2 232k . (148)-1 (two 2k) two – 1 – 12k (five 2k ) O( T0 2 ) 6The very same divergent element (1 – )-1 that was noticed for the SC in Equation (129) seems inside the total axial flux. The validity of Equation (148) is restricted to massive z, but additionally to big T0 compared to the mass k -1 . The dependence of FA on z is shown in Figure 6a. At k = 0, the coordinate-independence of FA is confirmed as well as the agreement together with the result in Equation (148) is exceptional even for small temperature, T0 = 0.5. At finite mass (k = 1), the flux FA decreases like z-k . The leading-order coefficient is temperature-dependent and also the result in Equation (148) becomes inaccurate at smaller temperatures ( T0 = 0.five k = 1).one hundred k = 0, T0 = 0.5 k = 1, T0 = 0.five k = 0, T0 = two k = 1, T0 =Analytical10– )FA ( z10-1 (10-10-(a)( = 0.9) 101 102z 102 1010.05 0.04 0.03 2 an – )[FA – FA ]z =0 0.02 0.01 0 -0.01 -0.02 -0.03 -0.(c)- )FA (= 0) z10-1 10-2 10-3 10-4 10-(b)1 (k k k k= 0, = 0, = 1, = 1,= 0.1 = 0.9 = 0.1 = 0.1 (Analyticalk = 0, = 0.1 k = 0, = 0.9 k = 1, = 0.1 k = 1, = 0.9 two an -(1 – )FA / 10010-10–0.05 10-TTFigure 6. (a) Dependence of FA on z for massless (k = 0) and massive (k = 1) quanta at small an ( T0 = 0.5) and huge ( T0 = two) temperatures. Temperature dependence of (b) FA ; and (c) FA – FAan for z = 0, normalised with respect to the prefactor /(1 – ). The analytical results FA at big z (panel a) and huge T0 (panels b,c) are offered in Equations (148) and (151), respectively.Symmetry 2021, 13,31 ofAt higher temperature, Equation (A12) may be employed to expand the hypergeometric function appearing in Equation (146):two Fk, 1 k; 1 2k; -(1 z two ) -sinhj 0=j (1 2k ) (1 z.